5.4
Conclusion
In this chapter, the counterintuitive fluorescence time trace of single photon emitters namely a single Ce ion in YAG crystal has been observed, which presents a gradual decay curve in the fluorescence time trace when it is under a CW laser excitation. A possible picture is drawn to explain this phenomenon, where deep level traps in the band gap are introduced as the donors to convert Ce back to Ce3+. So far this is only a proposed model, more experiments such as thermal luminescence or solid state spectroscopy, should be carefully performed to either support or optimize this model. Unfortunately, the observation indicates that the attempts of resonant excitation Ce ions by CW laser will be unsuccessful. However, we also notice that Ce shows good photostabil- ity under femtosecond pulsed laser irradiation. The studies suggest that the femtosecond laser pulses induce the two-photon absorption which restores the donor electrons in the deep level traps. Therefore, if the femtosecond pulsed laser and CW laser is applied si- multaneously, Ce shows good photostability. With the combination of low repetition rate femtosecond pulsed laser and CW laser offers a new way to excite Ce ions through its resonant optical transitions and coherent addressing single Ce spin qubit all-optically.
Chapter 6
All-optical addressing single Ce spin
qubits
In Chapter IV, high fidelity optical initialization, coherent manipulation and optical read- out of single Ce spin qubits have been demonstrated. It paves a way to study the inter- facing of the light and single rare earth solid-state qubits. Compared to the techniques for microwave manipulation as shown before, all-optical control is a fast and precise ap- proach to coherently manipulate single solid-state qubits. It is an essential step towards the development of integrated nanophotonic systems using single rare earth ions.
In Chapter V, the charge dynamics of single Ce ions has been studied. High peak intensity femtosecond laser pulses efficiently suppress the charge dynamics of Ce ions. It results in the good photostability of a Ce ion when it is excited under the combination of low repetition rate femtosecond pulsed laser and a CW laser. In this Chapter, high-resolution spectroscopy of single Ce ions in YAG is performed. Four narrow and spectral stable optical transitions can form two Λ system. Utilizing either of the Λ system, the preparation of coherent dark states of a single Ce spin qubit has been demonstrated. Exciting single Ce ions resonantly and addressing its spin states all-optically offers more degree of freedom to coherently control single Ce spin qubits.
6.1
Theory of coherent population trapping
Coherent population trapping(CPT) [155] and related physical phenomena (electromag- netically induced transparency [156, 157], slow light [158], lasing without inversion [159], etc.) as one of the common techniques in quantum optics have been discovered and developed in several decades [160]. It is a consequence of quantum interference effect between two excitation pathways when two ground levels are both excited to a common excited state. CPT was initially used in quantum ensembles to initialize, unitarily manipulate and readout the quantum states all-optically. Recent years, it has been demonstrated at single-ion level in atom systems and solids resulting in the all-optical addressing of single spin qubits [161–165], which is a key tool for precise and fast control of single qubits [166, 167].
74 6.1 Theory of coherent population trapping
Consider a three-level system interacting coherently with two external electromagnetic fields: probe and coupling field. The three-level Λ system is shown in Fig. 6.1(a), that |g1i
and |g2i are ground states, and they share a common excited state |ei. In the Λ system,
state |g1i and |ei are on resonance with frequency ωp = ωe− ωg1. While state |g2i and |ei
are on resonance with frequency ωc = ωe− ωg2. The frequency detuning ∆1 and ∆2 are
introduced from the resonant transitions ωp and ωcrespectively. Γeg1 and Γeg2 correspond
to the radiative decay rates from the excited state |ei to two ground states |g1i and |g2i.
Figure 6.1: (a) A three-level Λ system coherently interacts with two external electromagnetic fields. (b)-(f) Absorption spectrum with different parameters. (b) γg2g1 = 1, γeg1 = 3, and
Ωc= 0. (c) γg2g1 = 1, γeg1 = 3, and Ωc= 1. (d) γg2g1 = 1, γeg1 = 3, and Ωc = 2. (e) γg2g1 = 1,
γeg1 = 3, and Ωc= 4. (f) γg2g1 = 0.03, γeg1 = 3, and Ωc= 1.
When the three-level system is interacting with the external electromagnetic field, the CPT Hamiltonian is described by:
HCP T = H0+ H1. (6.1)
In Eq. 6.1 H0is the Hamiltonian of the three-level system without external electromagnetic
fields. H1represents the perturbation induced by the applied external field which is defined
by:
H1 = −qE · r, (6.2)
here q is the charge of the electron. E and r represent electric field and a position vector. With the rotation wave approximation, the HCPT is written as [160]:
HCPT= 1 2~ 0 0 Ωp 0 −2(∆1− ∆2) Ωc Ωp Ωc −2∆1 , (6.3)
6 All-optical addressing single Ce spin qubits 75
where Ωp and Ωc represent the optical Rabi frequency in between the |g1i ↔ |ei and
|g2i ↔ |ei transitions caused by probe and coupling field.
The eigenstates of the HCP T are calculated as [160]:
|a+i = sin θ sin φ|g
1i + cos φ|ei + cos θ sin φ|g2i, (6.4)
|a−i = sin θ sin φ|g1i − cos φ|ei + cos θ cos φ|g2i, (6.5)
|a0i = cos θ|g 1i − sin θ|g2i, (6.6) in which θ = arctan(Ωp Ωc) and φ = 1 2arctan( √ Ω2 p+Ω2c ∆1 ).
Here, |a0i is particularly interesting to us, since it can be obtained from Eq. 6.6 that:
he|HCPT|a0i = 0. (6.7)
Equation6.7 suggests no contribution of state |ei to states |a0i. In this case, the population of the excited state is 0, indicating no absorption and fluorescence from the system. It results in state |a0i as the dark state. This is so called coherent population trapping.
If the probe field is much weaker than the coupling field, it yields to Ωp Ωc. It yields
the θ in Eq. 6.3 close to 0. The |a0i state turns to |a0i = |g
1i. In addition, as soon as
the detuning of the probe field ∆1 = 0 (probe field is on resonance with the transition
|g1i ↔ |ei), |a+i and |a−i are written as: |a+i = √12(|g1i + |g2i) and |a−i = √12(|g1i − |g2i).
These are normally described as dressed state picture.
Followed by the master equation for the atomic density operator, the first order suscep- tibility χ(1) is given by [160]: χ(1)(−ωp, ωp) = |µ|2% 0~ × 4δ(|Ω c|2− 4δ∆1) − 4∆1γg22g1 ||Ωc|2+ (γeg1 + i2∆1)(γg2g1 + i2δ)|2 + i 8δ 2γ eg1 + 2γg1g2(|Ωc| 2+ γ g2g1γeg1 ||Ωc|2+ (γeg1 + i2∆1)(γg2g1 + i2δ)|2 (6.8) where γeg1, γg2g1 represent to the coherent decay rates from |ei to |g1i and from |g2i to
|g1i respectively. δ is the two photon detuning: δ = ∆1− ∆2. When the probe and pump
field are both on resonance with the optical transitions δ = ∆1 = 0, three parameters
γg2g1, Ωc and γeg1 remain important. In Eq. 6.8 the imaginary part of the susceptibility
Im[χ(1)] represents to the dissipation of the field by the system [160], which is represented
by the PLE spectrum of the three-level system.
The simulation of the Im[χ(1)] corresponds to different ratio between γ
g2g1, γeg1 and Ωc
is shown in Fig. 6.1(b)-(f). If the coupling field is zero (Fig. 6.1(b)), the PLE spectrum presents a lorentz like line shape, where the linewidth is determined by the coherence decay rate of the excited state γeg1. When the coupling field is applied, a dip appears
when the probe field frequency is on resonance with ωeg1. It indicates the successful
preparation of coherent dark states in between the ground states |g1i and |g2i. The width
76 6.1 Theory of coherent population trapping
the coupling power is heightened, the dip at the middle of the PLE spectrum goes deeper as shown in Fig. 6.1(c)-(e). The width of the dip is broadened if the coupling field is strong, which is known as power broadening effect. If the coupling field is weak and the coherence decay rate of the ground state is relatively long as displaced in Fig. 6.1(f), PLE spectrum of the CPT process expresses a narrow and sharp dip going down to the background level.
In the case of single Ce ions embedded in YAG, to generate the coherent dark states of single Ce spin qubits, two requirements should be under consideration.
(1) The construction of the Λ system. It is introduced in Chapter 3 if the external magnetic field is perpendicular to the laser beam direction, both spin-flip and non-flip transitions are allowed. To achieve this condition, we set-up an external magnetic field (see Appendix). The external magnetic field ( ~B ∼450 G) dominates the optical selection rules of Ce:YAG, indicating the permission of four optical transitions between the lowest Kramers doublets of the ground state and of the excited state, as presented in Fig. 6.2, where the Λ system can be formed.
Figure 6.2: The energy level and the selection rules of the optical transitions of single Ce ions in YAG.
(2) Acquirement of resonant transitions of single Ce ions. It is essential to resonant excite single Ce ions through the optical transitions to achieve the all-optical addressing of single electron spin qubits. On top of that, if the transition linewidth is broad (bigger than the ground state splitting) or presents huge spectral diffusion, the CPT signal will be almost invisible. The obtainment the ZPLs of single Ce ions and determine the resonant transition quality by means of photoluminescence excitation(PLE) spectrum is the next step.
6 All-optical addressing single Ce spin qubits 77